Conditional extremals in complete Riemannian manifolds

Conditional extremal curves in a complete Riemannian manifold MM are defined as the critical points of the squared L2L2 distance between the tangent vector field of a curve and a so-called prior   vector field. We prove that this L2L2 distance satisfies the Palais–Smale condition on the space of absolutely continuous curves joining two submanifolds of MM, and thus establish the existence of critical points. We also prove a Morse index theorem in the case where the two submanifolds are single points, and use the Morse inequalities to place lower bounds on the number of critical points of each index.